Honorable

I’m thinking about upgrading my graphics card, as my current one can’t run current games in HD (1920x1200) with high settings. I need a card that can run most current games in high or ultra at 1920x1200, 60 fps. This would be a 1060 or an RX 580. The catch is, while I game, my wife wants to watch a movie on the HD projector (1080p). The most idiotic scenario possible (has happened) is all of the above plus a 720p stream of a football match running on the secondary monitor.
My cpu can easily handle this. My current graphics card can handle this. But I’m not even sure what to look for when upgrading. How taxing is a 1080p movie on a graphics card? And a 720p stream? All while HD gaming? Do I need more memory on the graphics card? I don’t even know what to google...

Estimable

I think a single 1080TI is more than fine to hold on that much and dont struggle between multiple tasks at the same time.
Keep in mind that ANY single card you add will run pretty hot since you will be demanding much to it.
So a powerful card with a nice cooling system such as 1080ti its the only solution i came to... Also, you will be connecting 3 outputs 2 displays and 1 projector. A 1080ti is needed for such a titanical task.
IF possible i would suggest you to purchase any GTX 1080ti that comes with liquid cooling on it such as the Hydro GFX GTX 1080 Ti Liquid Cooled Version. I know it worths a little bit more than a Standard 1080ti but you will demand it pretty hard. The tasks individually are very simple and very easy even for a 1060 but all 3 at the same time its quite a titanic task and you need to be prepared for. Also, this allows you to change games for an even more demanding one in case you get bored of the ones you are playing and go for a AAA title while Streaming and watching movies.

Honorable

Thank you both for the answers. @RobCrezz I’ll look into the VGA, I thought a passive DVI to VGA could do it, but now that you mention I’ll research this better. @Ramlethal is HD movie watching really that taxing on a video card? I know a 1080 ti would do the job, but I don’t want to get a card that is unnecessary for me. Anyway, I’ll look at the local prices, maybe the difference is not that huge. Thanks!